Binormal distribution

If you're not familiar with the (univariate) normal distribution, we suggest that you first report here.

The multivariate normal distribution is by far the most widely used distribution in classical modeling techniques (e.g. Discriminant Analysis), that often explicitely assume the data distributions to be multinormal.

When only two variables are involved, the multivariate normal distribution is called the bivariate normal distibution, or simply the binormal distribution. For example, the joint distribution of the height and weight of individuals in a relatively homogenous population is approximately binormal.

The binormal distribution is just a special case of the general multivariate normal distribution. Yet, it is useful to devote a special entry to this distribution for at least two reasons :

* First, it is quite often met in practical applications. The equations pertaining to the bivariate case are simpler than those of the general multivariate case, and are therefore to be prefered.

- Any result pertaining to the binormal distribution can be easily visualized (for example, see animation below).

- The equations describing the properties of the bivariate normal distribution are more complex than those describing the univariate normal distribution, but they are still quite manageable. Further on, when an arbitrary number of variables will be considered, these equations will become very cumbersome and will be advantageously replaced by matrix equations. Therefore, the binormal distribution offers the opportunity for a "soft" introduction to linear algebra by first establishing results as "ordinary equations", then translating these equations into matrix notation.

This animation is a first contact with the bivariate normal distribution. It displays two (univariate) normal distributions, represented by two gaussian curves (one is horizontal, the other one is vertical). The standard deviations of these distributions can be changed by using the two cursors against yellow backgrounds.

The correlation coefficient of the corresponding two normal variables can be changed with the cursor against a blue background.

The "Book of Animations" on your computer

* Click on "Go" and observe the spatial distribution of the observations drawn from a binormal distribution. All adjustments may be modified without interrupting the animation.

- When r is equal to +1 or -1, the observations line up on a straight line whose slope depends on the standard deviations of the marginal distributions.

- When r = 0, the binormal distribution has an oblong shape whose axes are parallel to the axes.

We first establish the elementary properties of the standard bivariate normal distribution. We have two independent standard normal variables X and Y, and consider all linear combinations V = aX + bY of these two variables. We establish the conditions on a and b for V to be also standard normal, and discover that there is one free parameter left : the correlation coefficient r of X and V. The joint distribution of X and V is then called "the standard bivariate normal distribution with correlation coefficient r ".

So it is in fact a family of distributions indexed by the correlation coefficient r. It is depicted by the above animation when both standard deviations are kept at their initial values (1), and the value of the correlation coefficient is used to run through the family.

* We calculate the general form of the standard binormal distribution and establish its elementary properties. We will discover that when r = 0, the two r.v. X and V are independent.

* The conditional distributions of the standard binormal distribution are also normal, and we establish their properties.

* We then show that under an appropriate rotation of the reference frame, the binormal distribution appears as the joint distribution of two independent normal random variables, whose properties we calculate.

* We finally show that a line of constant probability density is an ellipse, (called covariance ellipse, or error ellipse) and we calculate the lengths and orientations of its axes.

We mentioned that if the two normal components of a binormal distribution are uncorrelated, then they are also independent. The terms "uncorrelated" and "independent" are often associated when it comes to normal r.v., and it is a common idea that uncorrelated r.v. are independent.

In this Tutorial, we give two counter-exemples that dispell this misconception. For each one, we exhibit a pair (X, Y) of standard normal r.v. that are uncorrelated, yet that are not independent. This "uncorrelated yet not independent" combinatation is somewhat unsual for normal variables, and has just as unsual consequences :

* A bivariate distribution may have normal marginal distributions, yet not be binormal.

In the Tutorial, we give the correct version of the link between "uncorrelated" and "independent" for normal variables.

Univariate normal distribution
Correlation coefficient
Covariance matrix

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